Order of Accuracy for Finite Difference Method Backward Euler

[evinda]

Hello! (Wave)
We are given the boundary / intial value problem for the heat equation:

$\left\{\begin{matrix}
u_t(t,x)=u_{xx}(t,x), \ \ x \in [a,b], \ \ t \geq 0\\
u(0,x)=u_0(x), \ \ \forall x \in [a,b] \\
u(t,a)=u(t,b)=0, \ \ \forall t \geq 0
\end{matrix}\right.$

I have written a code to approximate the solution of the problem.

How do we calculate the order of accuracy of the finite difference method backward euler?

I have found the error $$E^n=\max_{1 \leq i \leq N_x+1}|u^n_i-u(t_n, x_i)|, n=1, \dots, N_t+1$$

Do we have to take different values for $N_x$ to find the order of accuracy? (Thinking)

 

[evinda]

I have tried the following:function [p1]=order_fin_dif_back_euler [u1, ex1]=finite_difference_backward - Pastebin.com

The first two arguments of the function [m]finite_difference_backward_euler[/m] stands for the interval $[a,b]$, the third is the number of subintervals of this interval, the fourth one is $T_f$ ($t \in [0,T_f]$) , the last argument is the number of subintervals of $[0,T_f]$.

For [m]number of subintervals of [a,b]=20[/m] and [m]number of subintervals of [0,T_f]=400[/m] I got that:
[m]p1 = 0.1008[/m]The order of accuracy should tend to $2$. Is there a mistake at my code? (Thinking)